MPPT for Photovoltaic System Using Adaptive Fuzzy Backstepping Sliding Mode Control

Renewable energy has drawn much attention in recent years due to the high demand for green energy resources. The importance of solar panels (solar energy system) is greater nowadays as renewable sources since they exhibit many merits such as producing clean electrical energy, little maintenance and unlike other sources of renewable energy it has no geographical restrictions.

Solar PV systems have complex configurations that consist of components with nonlinear behaviors mainly due to the weather varying conditions such as temperature, irradiance and among others. These operating conditions are changed significantly with time which requires an adaptive control scheme to maintain adequate maximum power production in a practical operating environment.

Therefore, ensuring maximum electrical power extraction from photovoltaic systems, regardless of load changes and environmental conditions, is the primary objective control strategy, known as the Maximum Power Point Tracking Problem (MPPT) [1].

Many methods have been developed to determine the maximum power point (MPP) under all conditions [2-8]. There are numerous approaches, some of which are based on the well-known perturb and observe (P&O) concept [2, 3], others on the sliding mode control method [4, 5], artificial neural networks or fuzzy based algorithms [6, 7], and synergetic control [1, 7, 8].

Maximum power voltage (MPV)-based techniques using a two-loop MPPT control system are proposed in Refs. [9-11]. The first loop determines the PV array's MPV reference, while the second regulates the PV array's voltage to the reference voltage. The MPV reference search and PV voltage tracking are repeated until the maximum power is obtained. A hybrid technique consisting of two loops is presented in Ref. [12] to track MPP more effectively. MPP is estimated using an incremental conductance approach in the first loop. To regulate the system to the searched reference MPP, a second loop terminal sliding mode controller is constructed. Whereas Dahech et al. [13] proposed backstepping sliding mode control (BSMC) for the second loop. However, the main drawback of the SMC is the chattering. We propose in this paper to approximate discontinuous control using an adaptive fuzzy system based on the universal approximation theorem to tackle the problem of chattering mentioned in Ref. [13]. To ensure the system's global stability, the parameter of the fuzzy system is modified using an adaptation law based on the Lyapunov synthesis.

The goal of this work is to develop an adaptive fuzzy Backstepping sliding mode controller (AFBSMC) for MPPT, in order to overcome the problem of chattering. The whole system is modeled and simulated in Matlab/Simulink. Simulation results prove and confirm the effectiveness of the proposed approach in the elimination of the chattering.

The performance of this approach proves a high efficiency compared to the BSMC. The remainder of this paper is organized as follows. In Section 2, the MPPT system modeling is presented. The design of AFBSMC is exposed in Section 3. Section 4 uses numerical simulations to demonstrate the controller's usefulness in tracking MPP. The paper comes to a close with a conclusion.

To achieve MPP under the changing atmosphere, the Figure 5 illustrates the overall control structure. Here, I_pv and V_pv are measured from PV array and sent to the MPP searching algorithm, which generates the reference maximum power voltage $V_{\text {ref }}$. Then, the reference voltage $V_{\text {ref }}$ is given to the maximum power voltage based AFBSM controller for the maximum power tracking.

5.png

Figure 5. Proposed system

3.1 MPP searching algorithm

To achieve the maximum power operation, we use an incremental conductance method to search the MPP voltage $V_{\text {ref }}$.

The power slope $d P_{p v} / d V_{p v}$ can be expressed as:

$\frac{d P_{p v}}{d V_{p v}}=I_{p v}+V_{p v} \frac{d I_{p v}}{d V_{p v}}$

When the power slope $\frac{d P_{p v}}{d V_{p v}}=0$, i.e., $\frac{d i_{p v}}{d V_{p v}}=-\frac{I_{p v}}{V_{p v}}$, the PV system operates at the maximum power generation.

Therefore, the update law for $V_{\text {ref }}$ is given by the following rules [12]:

$\left\{\begin{array}{l}V_{r e f}=V_{r e f}(k-1)+\Delta V, \text { for } \frac{d i_{p v}}{d V_{p v}}>-\frac{I_{p v}}{V_{p v}} \\ V_{r e f}=V_{r e f}(k-1)-\Delta V, \text { for } \frac{d i_{p v}}{d V_{p v}}<-\frac{I_{p v}}{V_{p v}}\end{array}\right.$

3.2 Backstepping sliding mode controller

To extract the maximum power from a PV panel a backstepping sliding mode controller is designed. Where the objective of this is to let the panel PV voltage $V_{p v}$ track the reference maximum power voltage $V_{\text {ref }}$ by acting on the duty cycle $d(t)$ of the boost converter switch.

The recursive nature of the propose control design is similar to the standard Backstepping methodology. However, the proposed control design uses Backstepping to design controllers with a zero-order sliding surface at the last step [13]. The design proceeds as follows:

For the first step we consider zero-order sliding surface represented by following equation:

$e_{1}=x_{1}-x_{d}$     (7)

where: $x_{d}=V_{\text {ref }}$.

Considering an auxiliary tracking error variable:

$e_{2}=\dot{e}_{1}+\alpha_{1}$     (8)

Let the first Lyapunov function candidate:

$V_{1}=\left(\frac{1}{2}\right) e_{1}^{2}$     (9)

The time derivation of Eq. (5) is given by the Eq. (10):

$\dot{V}_{1}=e_{1} \dot{e}_{1}=e_{1}\left(e_{2}-\alpha_{1}\right)=-\lambda_{1} e_{1}^{2}+e_{1} e_{2}$     (10)

The stabilization of $\mathrm{e}_{1}$ can be obtained by introducing a new virtual control $\alpha_{1}$ Eq. (11), such that:

$\alpha_{1}=\lambda_{1} e_{1}, \lambda_{1}>0$    (11)

where $\lambda_{1}$ a positive feedback gain, such that $\alpha_{1}$ has been chosen in order to eliminate the non linearity and getting $\dot{V}_{1}(s)<0$. The term $e_{1} e_{2}$ of $\dot{V}_{1}$ will be eliminated in the next step, so the first sub system is stabilized.

For the second step we consider the following sliding surface:

$s=\lambda_{2} e_{1}-e_{2}$    (12)

The augmented Lyapunov function is given by Eq. (13):

$V_{2}=V_{1}+\left(\frac{1}{2}\right) s^{2}$     (13)

Eqns. (14)-(15) show the time derivative of $V_{2}$:

$\dot{V}_{2}=\dot{V}_{1}+s . \dot{s}$     (14)

$\dot{V}_{2}=-\lambda_{1} e_{1}^{2}+e_{1} e_{2}+s .\left[\lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)-\dot{e}_{2}\right]$    (15)

The Eqns. (16)-(18) represent the time derivative of $e_{2}$:

$\dot{e}_{2}=\ddot{e}_{1}+\dot{\alpha}_{1}$     (16)

$\dot{e}_{2}=\ddot{V}_{p v}-\ddot{x}_{1 d}+\dot{\alpha}_{1}$    (17)

$\dot{e}_{2}=-\frac{1}{C_{1}}\left[f_{1}(x)+g_{1}(x) d(t)\right]+\frac{1}{C_{1}} \dot{I}_{p v}-\ddot{x}_{1 d}+\dot{\alpha}_{1}$    (18)

One can obtain the Eq. (19):

$\dot{s}=\left[\lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)+\frac{1}{C_{1}}\left[f_{1}(x)+g_{1}(x) d(t)\right]\right.$

$\left.-\frac{1}{C_{1}} I_{p v}+\ddot{x}_{1 d}-\dot{\alpha}_{1}\right]$     (19)

We impose the following dynamic to the sliding surface:

$\dot{s}=-h \cdot(s+\beta \operatorname{sgn}(s))$    (20)

$\operatorname{sign}(.)$  is the usual sign function., where $h>0$ and $\beta>0$ .

Substituting Eq. (18) into Eq. (15) we obtain:

$\begin{aligned} \dot{V}_{2}=-\lambda_{1} \mathrm{e}_{1}^{2}+\mathrm{e}_{1} \mathrm{e}_{2} & \\+& \mathrm{s}\left[\lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)\right.\\ &-\left(-\frac{1}{C_{1}}\left[f_{1}(x)+g_{1}(x) d(t)\right]\right.\\ &\left.\left.+\frac{1}{C_{1}} I_{p v}-\ddot{x}_{1 d}+\dot{\alpha}_{1}\right)\right] \end{aligned}$     (21)

The control law is defined as:

$d(t)=d_{e q}(t)+\frac{C_{1}}{g_{1}(x)} d_{s w}(t)$    (22)

$d_{s w}(t)=-h(s+\beta \operatorname{sgn}(s))$     (23)

where, $d_{s w}(t)$ in the Eq. (23) is the switching control.

So we have:

$d(t)=\frac{1}{g_{1}(x)}\left[-f_{1}(x)-C_{1} \lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)+I_{p v}\right.$$-C_{1} \ddot{x}_{1 d}+C_{1} \dot{\alpha}_{1}-C_{1} h(\mathrm{~s}$$+\beta s g n(s)]$     (24)

The Eq. (15) is developed to:

$\dot{V}_{2}=-\lambda_{1} e_{1}^{2}+e_{1} e_{2}-s[h(\mathrm{~s}+\beta \operatorname{sgn}(s)]$     (25)

Introducing the norm in the Eq. (25), we get the Eqns. (26) and (27):

$\dot{V}_{2} \leq-\lambda_{1} e_{1}^{2}+e_{1} e_{2}-h s^{2}-\beta|s|$    (26)

$\dot{V}_{2} \leq-e^{T} p e-\beta|s|$    (27)

where, $e=\left[\begin{array}{ll}e_{1} & e_{2}\end{array}\right]^{T}$and P is a symmetric matrix defined as:

$P=\left[\begin{array}{cc}\lambda_{1}+h \lambda_{2}^{2} & -h \lambda_{2}-\frac{1}{2} \\ -h \lambda_{2}-\frac{1}{2} & h\end{array}\right]$

This proves the decreasing of Lyapunov function, which ensures that the closed-loop system is stable and robust.

This kind of sliding mode is certainly robust and stabilized but has two major drawbacks. The first lies in the presence of the sign function, where the control signal causes the phenomenon of chattering. The second disadvantage lies in the difficulty of the calculation of the constant $\beta, h$. To overcome these drawbacks several solutions have been presented in literature [15-17].

In order to resolve this problem, we propose to modify the following in the previous control law by using a fuzzy adaptive system. By letting the sliding area has input to approximate the term $d_{s w}(t)$. The fuzzy kind of lathing allows elimination of the phenomenon of chattering perfectly. At the same time as the adaptive appearance is designed to approximate the constant.

3.3 AFBSM controller of PV system

In this work we use the following If-Then rules [18, 19] to construct the fuzzy logic system:

$R_{i}:$ If $x_{1}$ is $F_{1}^{i}$ and ... and $x_{n}$ is $F_{n}^{i}$ then $y$ is $B^{i}, i=1,2, \ldots, n$

The fuzzy logic system with the singleton fuzzifier, product inference and center average defuzzifier is written as follows:

$y(x)=\frac{\sum_{i=1}^{n} \theta_{i} \prod_{j=1}^{n} \mu_{F_{j}^{i}\left(x_{j}\right)}}{\sum_{i=1}^{n}\left[\prod_{j=1}^{n} \mu_{F_{j}^{i}}\left(x_{j}\right)\right]}$     (28)

where,

$x=\left[x_{1}, \ldots, x_{n}\right]^{T} \in R^{n}, \mu_{F_{j}^{i}}\left(x_{j}\right)$ is the membership of $F_{j}^{i}$

$\theta_{i}=\max _{y \in R} \mu_{B^{i}}(y)$, let:

$\xi_{i}(x)=\frac{\prod_{j=1}^{n} \mu_{F_{j}^{i}}\left(x_{j}\right)}{\sum_{i=1}^{n}\left[\prod_{j=1}^{n} \mu_{F_{j}^{i}}\left(x_{j}\right)\right]}$    (29)

$\xi(x)=\left[\xi_{1}(x), \xi_{2}(x), \ldots, \xi_{n}(x)\right]^{T}$ and $\theta=\left[\theta_{1}, \theta_{2}, \ldots, \theta_{n}\right]^{T}$.

Then the fuzzy logic system can be rewritten by the Eq. (30):

$y(x)=\theta^{T} \xi(x)$     (30)

The following Lemma, points out that the above fuzzy logic systems are capable to uniformly approximating any continuous nonlinear function, over a compact set Ω_x. 

Lemma: [18-19].

For any given continuous function f(x) on a compact set $\Omega_{x} \subset R^{n}$; there exists a fuzzy logic system y(x)  In the form (30), such that for any given positive constant ε.$\sup _{x \in \Omega_{x}}|f(x)-y(x)| \leq \varepsilon$.

Then we propose to use a fuzzy system in the form (30) which can approximate the discontinuous control $d_{s w}(x)$, this latter is modeled by fuzzy system $\hat{h}(s)$. Then we have the following Eqns. (31) and (32):

$d_{s w}(x)=\hat{z}(s)+\Delta z(x)$     (31)

$\hat{z}(s)=\hat{\theta}_{z}^{T} \xi_{z}(s)$    （32）

Such that: w= Δz(x), is the approximation error.

So the control law is:

$d(t)=\frac{1}{g_{1}(x)}\left[-f_{1}(x)-C_{1} \lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)+\dot{I}_{p v}\right.$

$\left.-C_{1} \ddot{x}_{1 d}+C_{1} \dot{\alpha}_{1}-C_{1} \hat{z}(s)\right]$     (33)

The augmented Lyapunov function is given by Eq. (34):

$V_{2}=V_{1}+\left(\frac{1}{2}\right) s^{2}+\frac{1}{2 \eta_{z}} \tilde{\theta}_{z}^{T} \tilde{\theta}_{z}$     (34)

The derivative of this latter introducing Eq. (31), is given by Eqns. (35) and (36):

$\dot{V}_{2}=-\lambda_{1} e_{1}^{2}+e_{1} e_{2}+s \cdot\left[\lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)-(-\frac{1}{C 1}\left[f_{1}(x)+g_{1}(x)\left(d_{e q}(t)\right)\right]+\tilde{\theta}_{z} \xi(s) \underbrace{-\hat{z}(s)}+w+\frac{1}{C 1} \dot{I}_{p v}-\ddot{x}_{1 d}+\dot{\alpha}_{1})\right]-\frac{1}{\eta_{h}} \tilde{\theta}_{z}^{\tau} \hat{\theta}_{z}$      (35)

$\dot{V}_{2}=-\lambda_{1} e_{1}^{2}+e_{1} e_{2}+s \cdot\left[\lambda_{2}\left(e_{2}-\lambda_{1} e_{1}\right)-(-\frac{1}{\mathrm{C} 1}\left[f_{1}(x)+g_{1}(x)\left(d_{e q}(t)\right)\right] \underbrace{-\hat{z}(s)}+w+\frac{1}{C 1} \dot{I}_{p v}-\ddot{x}_{1 d}+\dot{\alpha}_{1})\right]-\frac{1}{\eta_{h}} \tilde{\theta}_{z}^{T}\left[\dot{\hat{\theta}}_{z}-\eta_{z} s \cdot \xi(s)\right]$     (36)

Choosing the adaptive law as follow:

$\dot{\hat{\theta}}_{z}=\eta_{z} s \xi(s)$    (37)

The optimal value of $\hat{z}(s)$ is such that:

$\left|\hat{z}^{*}(s)\right| \geq|w|$      (38)

The Eq. (34) is developed to:

$\dot{V}_{2} \leq=-\lambda_{1} e_{1}^{2}+e_{1} e_{2}+|s|\left[-\left|\hat{z}^{*}(s)\right|+|w|\right]$     (39)

$\dot{V}_{2} \leq-e^{T} Q e-\varphi|s| \rightarrow \dot{V}_{2} \leq 0$      (40)

where,

$\varphi=\left|\hat{z}^{*}(s)\right|-|w|$.

$e=\left[\begin{array}{ll}e_{1} & e_{2}\end{array}\right]^{T}$ and Q is a symmetric matrix with the following form:

$Q=\left[\begin{array}{cc}\lambda_{1}^{2} & -\frac{1}{2} \\ -\frac{1}{2} & 0\end{array}\right]$

This proves the decreasing of Lyapunov function, which ensures that the closed-loop system is stable and robust.